Subgroups — C₂×C₁₇×#64423 #23756218

Order 9996.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#3861	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#64423	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₇×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₇×#3861	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₇×C₁₇	C₂×C₃	1	2	yes	complemented
7	C₂²×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	maximal
8	C₂×C₁₇×AGL(1,𝔽₇)	—	7	1	no	maximal
9	C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₁₇×ΓL(1,𝔽₈)	—	7	2	no	—
11	C₂×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
12	C₂×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
13	C₂×C₁₇×ΓL(1,𝔽₈)	—	14	3	no	—
14	C₁₇×AGL(1,𝔽₇)	—	7	2	no	—
15	C₂×#64423	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₇×C₁₇	—	7	1	no	—
17	C₂×D₇×C₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
18	C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
19	C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
20	C₁₇×ΓL(1,𝔽₈)	—	14	3	no	—
21	C₂×#3861	C₂×C₁₇	1	1	yes	characteristic, complemented
22	#64423	C₂×C₁₇	1	2	yes	direct product factor
23	C₂×C₇×C₁₇	—	2	3	no	—
24	C₂×C₇×C₁₇	—	7	2	no	—
25	C₂×C₇×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
26	C₂×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	D₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
28	C₂²×C₃×C₁₇	—	49	1	no	Hall
29	C₂×C₇×D₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
30	#3861	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
31	C₇×C₁₇	—	2	3	no	—
32	C₇×C₁₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₂×C₃×C₁₇	—	49	2	no	—
35	C₂×C₃×C₁₇	—	49	1	no	—
36	C₂×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
37	C₇×D₇	C₂×C₃×C₁₇	1	2	yes	complemented
38	C₂²×ΓL(1,𝔽₈)	—	7	1	no	—
39	C₂×AGL(1,𝔽₇)	—	7	1	no	—
40	C₂²×C₁₇	—	7	1	no	Hall
41	C₃×C₁₇	—	49	1	no	Hall
42	C₇²	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
43	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
44	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
45	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
46	C₂×ΓL(1,𝔽₈)	—	14	3	no	—
47	AGL(1,𝔽₇)	—	7	2	no	—
48	C₂×C₁₇	—	7	2	no	—
49	C₂×C₁₇	#64423	1	1	yes	characteristic, direct product factor
50	C₂²×C₇	—	7	1	no	—
51	C₂×D₇	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
52	ΓL(1,𝔽₈)	—	7	1	no	—
53	ΓL(1,𝔽₈)	—	7	1	no	—
54	ΓL(1,𝔽₈)	—	14	3	no	—
55	C₁₇	C₂×#64423	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₇	—	2	3	no	—
57	C₂×C₇	—	7	2	no	—
58	C₂×C₇	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
59	C₂×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
60	D₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	2	yes	complemented
61	C₂²×C₃	—	49	1	no	Hall
62	C₇	—	2	3	no	—
63	C₇	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
64	C₇	C₂²×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
65	C₂×C₃	—	49	2	no	—
66	C₂×C₃	—	49	1	no	—
67	C₂²	—	7	1	no	Sylow
68	C₃	—	49	1	no	Sylow
69	C₂	—	7	2	no	—
70	C₂	C₁₇×#64423	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₇	#64423
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₇	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₁₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₁₇×#3861	C₂	semidirect
2. C₁₇×#64423	C₂	direct (N × H)
3. C₂×C₇×D₇×C₁₇	C₃	semidirect
4. C₁₇×#3861	C₂²	semidirect
5. C₂×C₇²×C₁₇	C₂×C₃	semidirect
6. C₇×D₇×C₁₇	C₂×C₃	semidirect
9. C₇²×C₁₇	C₂²×C₃	semidirect
15. C₂×#64423	C₁₇	direct (N × H)
17. C₂×D₇×C₁₇	ΓL(1,𝔽₈)	semidirect
21. C₂×#3861	C₂×C₁₇	semidirect
22. #64423	C₂×C₁₇	direct (N × H)
25. C₂×C₇×C₁₇	AGL(1,𝔽₇)	semidirect
26. C₂×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
27. D₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
29. C₂×C₇×D₇	C₃×C₁₇	semidirect
30. #3861	C₂²×C₁₇	semidirect
32. C₇×C₁₇	C₂×AGL(1,𝔽₇)	semidirect
33. C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	semidirect
36. C₂×C₇²	C₂×C₃×C₁₇	semidirect
37. C₇×D₇	C₂×C₃×C₁₇	semidirect
42. C₇²	C₂²×C₃×C₁₇	semidirect
49. C₂×C₁₇	#64423	direct (N × H)
51. C₂×D₇	C₁₇×ΓL(1,𝔽₈)	semidirect
55. C₁₇	C₂×#64423	direct (N × H)
58. C₂×C₇	C₁₇×AGL(1,𝔽₇)	semidirect
59. C₂×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	semidirect
60. D₇	C₂×C₁₇×ΓL(1,𝔽₈)	semidirect
63. C₇	C₂×C₁₇×AGL(1,𝔽₇)	semidirect
64. C₇	C₂²×C₁₇×ΓL(1,𝔽₈)	semidirect
70. C₂	C₁₇×#64423	direct (N × H)

Subgroups — C2×C17×#64423 #23756218

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₁₇×#64423 #23756218